Absolute instabilities in eccentric Taylor–Couette–Poiseuille flow. (25th February 2014)

• Record Type: Journal Article
• Authors: Leclercq, Colin; Pier, Benoît; Scott, Julian F.
• Is Part Of: Journal of fluid mechanics. Volume 741(2014:Feb.)
• Page Start: 543
• Page End: 566
• Publication Date: 2014-02-25
• Subjects: Fluid mechanics -- Periodicals; 532.005
• DOI: 10.1017/jfm.2013.646
• Languages: English
• ISSNs: 0022-1120

Abstract

<abstract> <title>Abstract</title> <p>The effect of eccentricity on absolute instabilities (AI) in the Taylor–Couette system with pressure-driven axial flow and fixed outer cylinder is investigated. Five modes of instability are considered, characterized by a pseudo-angular order <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghjkz6h5c" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$m$]]></tex-math></alternatives></inline-formula>, with here <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghjkz6fm4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\vert m\vert \leq 2$]]></tex-math></alternatives></inline-formula>. These modes correspond to toroidal (<inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghjkz6jq4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$m=0$]]></tex-math></alternatives></inline-formula>) and helical structures (<inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghjkz6gzs" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$m\neq 0$]]></tex-math></alternatives></inline-formula>) deformed by the eccentricity. Throughout the parameter range, the mode with the largest absolute growth rate is always the Taylor-like vortex flow corresponding to <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghjkz6fb0" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$m=0$]]></tex-math></alternatives></inline-formula>. Axial advection, characterized by a Reynolds number <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghjkz6jgh" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[${\mathit{Re}_z}$]]></tex-math></alternatives></inline-formula>, carries perturbations downstream, and has a strong stabilizing effect on AI. On the other hand, the effect of the eccentricity <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghjkz6jcz" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$e$]]></tex-math></alternatives></inline-formula> is complex: increasing <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghjkz6hcg" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$e$]]></tex-math></alternatives></inline-formula> generally delays AI, except for a range of moderate eccentricites <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghjkz6hz8" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[${0.3\lesssim e \lesssim 0.6}$]]></tex-math></alternatives></inline-formula>, where it favours AI for large enough <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghjkz6g2b" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[${\mathit{Re}_z}$]]></tex-math></alternatives></inline-formula>. This striking behaviour is in contrast with temporal instability, always inhibited by eccentricity, and where left-handed helical modes of increasing <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghjkz6jdg" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\vert m\vert $]]></tex-math></alternatives></inline-formula> dominate for larger <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghjkz6mwp" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[${\mathit{Re}_z}$]]></tex-math></alternatives></inline-formula>. The instability mechanism of AI is clearly centrifugal, even for the larger values of <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghjkz6h0s" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[${\mathit{Re}_z}$]]></tex-math></alternatives></inline-formula> considered, as indicated by an energy analysis. For large enough <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghjkz6gtq" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[${\mathit{Re}_z}$]]></tex-math></alternatives></inline-formula>, critical modes localize in the wide gap for low <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghjkz6jzr" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$e$]]></tex-math></alternatives></inline-formula>, but their energy distribution is shifted towards the diverging section of the annulus for moderate <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghjkz6gwr" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$e$]]></tex-math></alternatives></inline-formula>. For highly eccentric geometries, AI are controlled by the minimal annular clearance, and the critical modes are confined to the vicinity of the inner cylinder. Untangling the AI properties of each <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pghjkz6n0q" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$m$]]></tex-math></alternatives></inline-formula> requires consideration of multiple pinch points.</p> </abstract>